On the subject of reiter cavalry La Noue had this to say:

Quote:

Herein we are to consider two things which experience hath confirmed. The one, that the Reiter are never so dangerous as when they be mingled with the enemy, for then be they all fire. The other, the two squadrons meeting, they have scarce discharged the second pistol but either the one or the other turneth away. For they contesteth no longer as the Romans did against other nations, who oftentimes keep the field fighting two hours face to face before either party turned back.

He claims that because of the danger of firearms, cavalry engagements now only lasted an extremely short amount of time. Not even long enough to discharge more than one pistol before the other side broke off. I'm wondering if there was a similar trend in the length of infantry engagements.

Pistols would have been less common among infantry but sleeves of musketeers were typically attached to infantry and at close range a musket would likely have had enough force to kill two or three men in a single shot. In a depiction of a push of pike from 1658 you can see the center being enveloped by gunsmoke:
http://imgur.com/a/9tqwo

If the melee lasted even a few minutes the rear ranks of musketeers would have had time to reload their weapons and pour continuous fire into the enemy flanks, turning the engagement into a bloodbath. Some writers in the late 16th century were already noting that it was becoming uncommon for battles to be decided by handblows. Maybe the growing presence of firearms was making it so that even when a "push of pike" did happen it only lasted a moment before one side broke and ran?

Not really. Infantry fights in the English Civil War could go on for hours before they were decided one way or the other. The thing is, cavalry couldn't just stop a short distance away and shoot at each other for long periods of time because that would deprive them of their main assets (i.e. their mobility and the momentum of their charge). Sooner or later one side would get worked up and charge, or the other would get unnerved and flee. Infantry, on the other hand, could come to a dead stop fifty to a hundred yards apart and go on merrily shooting at each other for hours if there was no hand-to-hand fighting involved (and even with hand-to-hand fighting, there could be an hours-long engagement made up of pulses of intense fighting lasting only a few minutes each interspersed by longer lulls in which both sides would withdraw to catch their collective breaths).

The dynamics seem to have changed by the 18th and 19th centuries, though. The Napoleonic Wars saw thousands of bayonet charges but very few bayonet fights (probably less than a hundred over nearly twenty years). Firefights could still take hours, but once one side decided to move in with the bayonet then the decision was usually achieved without any physical hand-to-hand contact (either the defenders would flee from the attack, or the attackers would waver in the face of defenders who wouldn't budge and then be forced to retreat, sometimes under the incentive of a countercharge).